Ising model: Difference between revisions

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The [[Metropolis–Hastings algorithm]] is the most commonly used Monte Carlo algorithm to calculate Ising model estimations.<ref name="Newman" /> The algorithm first chooses ''selection probabilities'' ''g''(μ, ν), which represent the probability that state ν is selected by the algorithm out of all states, given that one is in state μ. It then uses acceptance probabilities ''A''(μ, ν) so that [[detailed balance]] is satisfied. If the new state ν is accepted, then we move to that state and repeat with selecting a new state and deciding to accept it. If ν is not accepted then we stay in μ. This process is repeated until some stopping criterion is met, which for the Ising model is often when the lattice becomes [[ferromagnetic]], meaning all of the sites point in the same direction.<ref name="Newman" />

When implementing the algorithm, one must ensure that ''g''(μ, ν) is selected such that [[ergodicity]] is met. In [[thermal equilibrium]] a system's energy only fluctuates within a small range.<ref name="Newman" /> This is the motivation behind the concept of '''single-spin-flip dynamics''',<ref name="pre0">{{cite journal|url= http://journals.aps.org/pre/abstract/10.1103/PhysRevE.90.032141|title= M. Suzen "Effective ergodicity in single-spin-flip dynamics"|journal= Physical Review E|date= 29 September 2014|volume= 90|issue= 3|page= 032141|doi= 10.1103/PhysRevE.90.032141|language=en-US|access-date=2022-08-09|last1= Süzen|first1= Mehmet|pmid= 25314429|arxiv= 1405.4497|bibcode= 2014PhRvE..90c2141S|s2cid= 118355454}}</ref> which states that in each transition, we will only change one of the spin sites on the lattice.<ref name="Newman" /> Furthermore, by using single- spin-flip dynamics, one can get from any state to any other state by flipping each site that differs between the two states one at a time. The maximum amount of change between the energy of the present state, ''H''_μ and any possible new state's energy ''H''_ν (using single-spin-flip dynamics) is 2''J'' between the spin we choose to "flip" to move to the new state and that spin's neighbor.<ref name="Newman" /> Thus, in a 1D Ising model, where each site has two neighbors (left and right), the maximum difference in energy would be 4''J''. Let ''c'' represent the ''lattice coordination number''; the number of nearest neighbors that any lattice site has. We assume that all sites have the same number of neighbors due to [[periodic boundary conditions]].<ref name="Newman" /> It is important to note that the Metropolis–Hastings algorithm does not perform well around the critical point due to critical slowing down. Other techniques such as multigrid methods, Niedermayer's algorithm, [[Swendsen–Wang algorithm]], or the [[Wolff algorithm]] are required in order to resolve the model near the critical point; a requirement for determining the critical exponents of the system.

Specifically for the Ising model and using single-spin-flip dynamics, one can establish the following. Since there are ''L'' total sites on the lattice, using single-spin-flip as the only way we transition to another state, we can see that there are a total of ''L'' new states ν from our present state μ. The algorithm assumes that the selection probabilities are equal to the ''L'' states: ''g''(μ, ν) = 1/''L''. [[Detailed balance]] tells us that the following equation must hold:

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==See also==

{{div col|colwidth=25em}}

{{too many see alsos|date=November 2024}}

* [[ANNNI model]]

* [[Binder parameter]]

* [[Boltzmann machine]]

* [[Conformal bootstrap]]

* [[Construction of an irreducible Markov chain in the Ising model]]

* [[Geometrical frustration]]

* [[Geometrically frustrated magnet]]

* [[Heisenberg model (classical)|Classical Heisenberg model]]

* [[Heisenberg model (quantum)|Quantum Heisenberg model]]

* [[Hopfield net]]

* [[Ising critical exponents]]

* [[John Clive Ward|J. C. Ward]]

* [[Kuramoto model]]

* [[Maximal evenness]]

* [[Order operator]]

* [[PottsSpin model]] (common with [[Ashkin–Teller model]])

* [[Random cluster model]]

* [[Spin model]]s

* [[Swendsen–Wang algorithm]]

* [[t-J model]]

* [[Two-dimensionalClassical critical IsingXY model]]

* [[WolffZN algorithmmodel]]

{{div col end}}

* [[XY model]]

* [[Z N model]]

==Footnotes==

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==References==

{{Refbegin|30em}}

*{{Citation | last1=Barth | first1=P. F. |author-link1=Peter F. Barth | year=1981 | title= Cooperativity and the Transition Behavior of Large Neural Nets | pages=1–118 | journal= Master of Science Thesis | publisher= University of Vermont | ___location= Burlington |oclc=8231704 }}

*{{Citation | last1=Baxter | first1=Rodney J. | title=Exactly solved models in statistical mechanics | url=https://physics.anu.edu.au/theophys/baxter_book.php | publisher=Academic Press Inc. [Harcourt Brace Jovanovich Publishers] | ___location=London | isbn=978-0-12-083180-7 | mr=690578 | year=1982 }}

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* [http://ibiblio.org/e-notes/Perc/contents.htm Phase transitions on lattices]

* [http://www.sandia.gov/media/NewsRel/NR2000/ising.htm Three-dimensional proof for Ising Model impossible, Sandia researcher claims]

* [http://isingspinwebgl.com Interactive Monte Carlo simulation of the Ising, XY and Heisenberg models with 3D graphics (requires WebGL compatible browser)]

* [https://github.com/AmazaspShumik/BayesianML-MCMC/blob/master/Gibbs%20Ising%20Model/GibbsIsingModel.m Ising Model code ], [https://github.com/AmazaspShumik/BayesianML-MCMC/blob/master/Gibbs%20Ising%20Model/imageDenoisingExample.m image denoising example with Ising Model]

* [http://www.damtp.cam.ac.uk/user/tong/statphys/five.pdf David Tong's Lecture Notes ] provide a good introduction